Problem 18 - Algebra and Trigonometry

Question

Evaluate the function at the indicated values. $$ \begin{array}{l}{f(x)=x^{3}-4 x^{2}} \\ {f(0), f(1), f(-1), f\left(\frac{3}{2}\right), f\left(\frac{x}{2}\right), f\left(x^{2}\right)}\end{array} $$

Step-by-Step Solution

Answer

\(f(0) = 0, f(1) = -3, f(-1) = -5, f\left(\frac{3}{2}\right) = -\frac{9}{8}, f\left(\frac{x}{2}\right) = \frac{x^3}{8} - x^2, f(x^2) = x^6 - 4x^4\).

Step 1: Evaluate f(0)

To find \(f(0)\), substitute \(x = 0\) into the function \(f(x) = x^3 - 4x^2\). This gives: \(f(0) = 0^3 - 4(0)^2 = 0\).

Step 2: Evaluate f(1)

Substitute \(x = 1\) into the function. Thus, \(f(1) = 1^3 - 4(1)^2 = 1 - 4 = -3\).

Step 3: Evaluate f(-1)

Substitute \(x = -1\) into the function. Thus, \(f(-1) = (-1)^3 - 4(-1)^2 = -1 - 4 = -5\).

Step 4: Evaluate f\left(\frac{3}{2}\right)

Substitute \(x = \frac{3}{2}\) into the function. Calculate: \(f\left(\frac{3}{2}\right) = \left(\frac{3}{2}\right)^3 - 4\left(\frac{3}{2}\right)^2 = \frac{27}{8} - 4\left(\frac{9}{4}\right)\). This simplifies to \(\frac{27}{8} - \frac{36}{8} = -\frac{9}{8}\).

Step 5: Evaluate f\left(\frac{x}{2}\right)

Substitute \(x = \frac{x}{2}\) into the function, then simplify. We have: \(f\left(\frac{x}{2}\right) = \left(\frac{x}{2}\right)^3 - 4\left(\frac{x}{2}\right)^2\). This expands to: \(f\left(\frac{x}{2}\right) = \frac{x^3}{8} - \frac{4x^2}{4}\), resulting in: \(\frac{x^3}{8} - x^2\).

Step 6: Evaluate f(x^2)

Substitute \(x = x^2\) into the function and simplify: \(f(x^2) = (x^2)^3 - 4(x^2)^2\). This becomes \(x^6 - 4x^4\).

Key Concepts

Understanding Polynomial Functions

A polynomial function is a type of mathematical expression that involves a sum of powers of variables, each multiplied by a coefficient. The function equation is typically of the form: \( a_nx^n + a_{n-1}x^{n-1} + \ ... + a_1x + a_0 \), where \(a_n, a_{n-1},..., a_0\) are constants. These functions are quite common in algebra and calculus due to their wide applicability, especially in describing curves and modeling real-world phenomena.
In our exercise, the polynomial function given is \( f(x) = x^3 - 4x^2 \). This function consists of two terms: \( x^3 \) and \( -4x^2 \) (both are polynomial terms). The degree of the function is determined by the highest power of \( x \), which in this case is 3. This represents a cubic polynomial.
Key characteristics of polynomial functions include their smooth, continuous nature, and the fact that they can be added, subtracted, and multiplied to create new polynomial functions.

Substitution Method Explained

The substitution method in function evaluation involves replacing the variable \( x \) with specific values or expressions to find the corresponding output. This process helps transform polynomial functions into numerical values or simpler expressions, enabling the evaluation of the function for given inputs.
In terms of our specific exercise, substitution was used multiple times:

• For \( f(0) \), we substitute \( x = 0 \) to find the output as \( 0 \).
• In \( f(1) \), replacing \( x = 1 \) yields an output of \( -3 \).
• For \( f(-1) \), inserting \( x = -1 \) results in \( -5 \).
• When \( x = \frac{3}{2} \), the substitution yields \( -\frac{9}{8} \).
• Substituting \( x = \frac{x}{2} \) results in a new expression: \( \frac{x^3}{8} - x^2 \).
• Finally, for \( x = x^2 \), substitution gives \( x^6 - 4x^4 \).

Substitution provides a practical and effective way to test various scenarios within a polynomial function.

Exploring Function Simplification

Function simplification is a key step in evaluating functions after substitution. It involves performing algebraic operations to reduce expressions into their simplest form. This process makes the function easier to handle and interpret, removing unnecessary complexity and revealing insights into the behavior and characteristics of the function.
After substituting specific values or expressions for \( x \), simplification arises from using arithmetic operations like addition, subtraction, multiplication, and division to obtain a leaner form.
Take for example \( f\left(\frac{3}{2}\right) \): substitute and simplify from \( \frac{27}{8} - 4\left(\frac{9}{4}\right) \) to \( \frac{27}{8} - \frac{36}{8} = -\frac{9}{8} \). Similarly, when substituting \( x = \frac{x}{2} \), the operations result in the expression \( \frac{x^3}{8} - x^2 \). Each expression must be simplified to its base level for accurate evaluating and interpretation.

Decoding Algebraic Expressions

Algebraic expressions, like those in polynomial functions, consist of variables, coefficients, and operations arranged into terms. Understanding these expressions helps in evaluating functions as they define mathematical relationships involving constants and variables. In algebraic expressions, each term must follow specific operations: multiplication, division, addition, and subtraction.
Consider the algebraic expression given in the exercise: \( f(x) = x^3 - 4x^2 \). Here, the variables are \( x \), and the coefficients are implicit numerical values indicating the number by which each term is multiplied. The operations are power (shown by the exponents), subtraction, and multiplication.
Algebraic manipulation involves an understanding of these terms and operations, allowing for effective substitution and simplification during polynomial function evaluation. This understanding is key to handling more complex algebraic equations with confidence.